Question

What is a recursive case?

Short answer

Answer: A recursive case is the part of a recursive function that calls itself to solve a smaller instance of the original problem. It reduces the problem into simpler subproblems, eventually reaching the base case. The base case is a situation where the function stops calling itself and returns a result directly, preventing infinite recursion. The recursive case helps break down the problem until the base case is reached, which then solves the smaller instances to obtain the final result.

Step-by-step solution

Understand recursion

Recursion is a process by which a function is defined in terms of itself, often with a base case to stop infinite repetition. In recursion, problems are solved by breaking them into smaller, simpler instances of the same problem.

Define base case

A base case is a situation where a recursive function stops calling itself and returns a result directly. The base case is important because it prevents infinite recursion, which leads to runtime errors or program crashes.

Define recursive case

A recursive case is the part of the function that calls itself to solve a smaller instance of the original problem. The recursive call is designed to reduce the problem into simpler subproblems, eventually reaching the base case.

Example of recursion using the factorial function

The factorial function (denoted as n! in_mathematics) is a common example of recursion. It is defined as the product of all positive integers less than or equal to n. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120. Here's a factorial function with both base and recursive cases: ``` def factorial(n): # Base case if n == 0 or n == 1: return 1 # Recursive case else: return n * factorial(n - 1) ``` In the recursive case above (else part), the function calls itself with a smaller value (n - 1). When the base case is reached, the function stops calling itself and returns the result.

Analyze the steps of the recursion process in the example

Let's trace the steps of the recursion process when calculating the factorial of 3: 1. factorial(3) is called. 2. Since 3!=0 and 3!=1, the recursive case is executed: 3 * factorial(3 - 1) or 3 * factorial(2). 3. factorial(2) is called. 4. Since 2!=0 and 2!=1, the recursive case is executed: 2 * factorial(2 - 1) or 2 * factorial(1). 5. factorial(1) is called. 6. Since 1 == 1, the base case is executed: return 1. 7. factorial(2) = 2 * 1, so it returns 2. 8. factorial(3) = 3 * 2, so it returns 3 * factorial(2) = 3 * 2 = 6. This way, we can see how the recursion process breaks the problem down until the base case is reached and then solves the smaller instances to get the result.